**From Table 11 in HEI report (p. 28):**

**Al l Causes (MSA** 

**& DIFF): Mean estimate:**  **Lower bound estimate:** PM1 = 15 PM1 = 20 PM1 = 15 PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612 input: RR = 1.128 RR = 1.079 RR = 1.077

ln(RR) = 0.120446 ln(RR) = 0.076035 ln(RR) = 0.074179 gam ma = 0.109635 gam ma = 0.109695 gam ma = 0.067521

gam ma = 0.109665 gam ma = 0.06758

1.128 1.077 1.079 1.048 RR from standard Cox log-linear model for delta PM =10: 1.057 RR from random effects log-linear model for delta PM =10: 1.078

**CPD (MSA**

**& DIFF): Mean estimate:**  **Lower bound estimate:** PM1 = 15 PM1 = 20 PM1 = 15 PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612 input: RR = 1.208 RR = 1.127 RR = 1.132

ln(RR) = 0.188966 ln(RR) = 0.119559 ln(RR) = 0.123986 gam ma = 0.172004 gam ma = 0.172488 gam ma = 0.112857

gam ma = 0.172246 gam ma = 0.112612

1.208 1.132 1.127 1.081 RR from standard Cox log-linear model for delta PM =10: 1.138 RR from random effects log-linear model for delta PM =10: 1.128

**IHD (MSA**

**& DIFF): Mean estimate:**  **Lower bound estimate:** PM1 = 15 PM1 = 20 PM1 = 15 PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612 input: RR = 1.484 RR = 1.283 RR = 1.311

ln(RR) = 0.394741 ln(RR) = 0.249201 ln(RR) = 0.27079gam ma = 0.359309 gam ma = 0.359521 gam ma = 0.246484

gam ma = 0.359415 gam ma = 0.246294

1.484 1.311 1.283 1.186 RR from standard Cox log-linear model for delta PM =10: 1.242 RR from random effects log-linear model for delta PM =10: 1.287

**Lung Cancer (MSA**

**& DIFF): Mean estimate:**  **Lower bound estimate:** PM1 = 15 PM1 = 20 PM1 = 15 PM0 = 5 PM0 = 10 PM0 = 5

ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM0)) 1.609438 ln(PM0)) 2.302585 ln(PM0)) 1.609438

difference: 1.098612 difference: 0.693147 difference: 1.098612 input: RR = 1.236 RR = 1.143 RR = 1.114

ln(RR) = 0.21188 ln(RR) = 0.133656 ln(RR) = 0.107957 gam ma = 0.192862 gam ma = 0.192825 gam ma = 0.098267

gam ma = 0.192844 gam ma = 0.098613

1.236 1.114 1.143 1.071**Lower bound estimate: Upper bound estimate:**  PM1 = 20 PM1 = 15 PM1 = 20 PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147 RR = 1.048 RR = 1.183 RR = 1.112 ln(RR) = 0.046884 ln(RR) = 0.168054 ln(RR) = 0.10616 gam ma = 0.067639 gam ma = 0.152969 gam ma = 0.153157

gam ma = 0.153063

1.183 1.112

**Lower bound estimate: Upper bound estimate:**  PM1 = 20 PM1 = 15 PM1 = 20 PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147 RR = 1.081 RR = 1.29 RR = 1.174 ln(RR) = 0.077887 ln(RR) = 0.254642 ln(RR) = 0.160417 gam ma = 0.112367 gam ma = 0.231785 gam ma = 0.231432

gam ma = 0.231609

1.290 1.174

**Lower bound estimate: Upper bound estimate:**  PM1 = 20 PM1 = 15 PM1 = 20 PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147 RR = 1.186 RR = 1.68 RR = 1.387 ln(RR) = 0.170586 ln(RR) = 0.518794 ln(RR) = 0.327143gam ma = 0.246104 gam ma = 0.472226 gam ma = 0.471968

gam ma = 0.472097

1.680 1.387

**Lower bound estimate: Upper bound estimate:**  PM1 = 20 PM1 = 15 PM1 = 20 PM0 = 10 PM0 = 5 PM0 = 10

ln(PM1)) 2.995732 ln(PM1)) 2.70805 ln(PM1)) 2.995732 ln(PM0)) 2.302585 ln(PM0)) 1.609438 ln(PM0)) 2.302585

difference: 0.693147 difference: 1.098612 difference: 0.693147 RR = 1.071 RR = 1.372 RR = 1.221 ln(RR) = 0.068593 ln(RR) = 0.31627 ln(RR) = 0.19967 gam ma = 0.098958 gam ma = 0.287881 gam ma = 0.288063

gam ma = 0.287972

1.372 1.2210

0.05 0.1 0.15 0.2 0.25 0.3

0 10 20 30 40 50 60**E s** **t i** **m a** **t e** **d A** **l l** **C a** **u s** **e M** **o r** **t a** **l i** **t y** **A s** **s o** **c i** **a t** **e d** **w i** **t h** **P M** **2 .** **5 O** **v e** **r L** **M L** 

**PM 2.5 Concentration**  random effects log-linear model random effects log-log model**Log-log vs. log linear model -- Krewski et al. (2009), later period -- all cause mortality**

beta from log-linear model 0.007511 (derived from HR for linear PM from Table 11) beta from log-log model 0.109665 (derived from HRs for log(PM) from Table 11)

PM LN(PM) LML LN(LML) PM - LML LN(PM) - LN(LML) log-linear log-log

6 1.791759 5.8 1.757858 0.2 0.033901552 0.001501 0.003711 7 1.94591 1.2 0.188052232 0.008972 0.020412 8 2.079442 2.2 0.321583624 0.016388 0.034652 9 2.197225 3.2 0.43936666 0.023748 0.047041 10 2.302585 4.2 0.544727175 0.031053 0.057988 11 2.397895 5.2 0.640037355 0.038303 0.067783 12 2.484907 6.2 0.727048732 0.045499 0.076636 13 2.564949 7.2 0.80709144 0.052641 0.084706 14 2.639057 8.2 0.881199412 0.05973 0.092114 15 2.70805 9.2 0.950192284 0.066766 0.098957 16 2.772589 10.2 1.014730805 0.073749 0.105312 17 2.833213 11.2 1.075355427 0.080679 0.111241 18 2.890372 12.2 1.13251384 0.087558 0.116794 19 2.944439 13.2 1.186581062 0.094386 0.122015 20 2.995732 14.2 1.237874356 0.101162 0.12694 21 3.044522 15.2 1.28666452 0.107888 0.131599 22 3.091042 16.2 1.333184536 0.114563 0.136018 23 3.135494 17.2 1.377636298 0.121189 0.14022 24 3.178054 18.2 1.420195913 0.127764 0.144223 25 3.218876 19.2 1.461017907 0.134291 0.148046 26 3.258097 20.2 1.50023862 0.140769 0.151702 27 3.295837 21.2 1.537978948 0.147198 0.155206 28 3.332205 22.2 1.574346593 0.153579 0.158568 29 3.367296 23.2 1.609437912 0.159913 0.1618 30 3.401197 24.2 1.643339464 0.166199 0.164911 31 3.433987 25.2 1.676129287 0.172438 0.167908 32 3.465736 26.2 1.707877985 0.17863 0.1708 33 3.496508 27.2 1.738649644 0.184776 0.173594 34 3.526361 28.2 1.768502607 0.190876 0.176295 35 3.555348 29.2 1.797490144 0.19693 0.178909 36 3.583519 30.2 1.825661021 0.202939 0.181442 37 3.610918 31.2 1.853059995 0.208903 0.183898 38 3.637586 32.2 1.879728242 0.214823 0.186281 39 3.663562 33.2 1.905703729 0.220698 0.188596 40 3.688879 34.2 1.931021537 0.226529 0.190845 41 3.713572 35.2 1.955714149 0.232317 0.193033 42 3.73767 36.2 1.979811701 0.238061 0.195163 43 3.7612 37.2 2.003342198 0.243762 0.197237 44 3.78419 38.2 2.026331716 0.249421 0.199259 45 3.806662 39.2 2.048804572 0.255037 0.20123 46 3.828641 40.2 2.070783479 0.260612 0.203153 47 3.850148 41.2 2.092289684 0.266144 0.20503 48 3.871201 42.2 2.113343093 0.271635 0.20686349 3.89182 43.2 2.133962381 0.277085 0.208655 50 3.912023 44.2 2.154165088 0.282495 0.2104060

0.05 0.1 0.15 0.2 0.25 0.3

0 10 20 30 40 50 60**E s** **t i** **m a** **t e** **d A** **l l** **C a** **u s** **e M** **o r** **t a** **l i** **t y** **A s** **s o** **c i** **a t** **e d** **w i** **t h** **P M** **2 .** **5 O** **v e** **r L** **M L** 

**PM 2.5 Concentration**  standard log-linear model random effects log-log model**Log-log vs. log linear model -- Krewski et al. (2009), later period -- all cause mortality**

beta from log-linear model 0.005543 (derived from HR for standard Cox proportional hazard model from Table 33) beta from log-log model 0.109665 (derived from HRs for log(PM) from Table 11)

PM LN(PM) LML LN(LML) PM - LML LN(PM) - LN(LML) log-linear log-log

6 1.791759 5.8 1.757858 0.2 0.033901552 0.001108 0.003711 7 1.94591 1.2 0.188052232 0.00663 0.020412 8 2.079442 2.2 0.321583624 0.012122 0.034652 9 2.197225 3.2 0.43936666 0.017583 0.047041 10 2.302585 4.2 0.544727175 0.023014 0.057988 11 2.397895 5.2 0.640037355 0.028415 0.067783 12 2.484907 6.2 0.727048732 0.033786 0.076636 13 2.564949 7.2 0.80709144 0.039127 0.084706 14 2.639057 8.2 0.881199412 0.044439 0.092114 15 2.70805 9.2 0.950192284 0.049721 0.098957 16 2.772589 10.2 1.014730805 0.054975 0.105312 17 2.833213 11.2 1.075355427 0.060199 0.111241 18 2.890372 12.2 1.13251384 0.065394 0.116794 19 2.944439 13.2 1.186581062 0.070561 0.122015 20 2.995732 14.2 1.237874356 0.075699 0.12694 21 3.044522 15.2 1.28666452 0.080808 0.131599 22 3.091042 16.2 1.333184536 0.08589 0.136018 23 3.135494 17.2 1.377636298 0.090943 0.14022 24 3.178054 18.2 1.420195913 0.095969 0.144223 25 3.218876 19.2 1.461017907 0.100966 0.148046 26 3.258097 20.2 1.50023862 0.105936 0.151702 27 3.295837 21.2 1.537978948 0.110879 0.155206 28 3.332205 22.2 1.574346593 0.115794 0.158568 29 3.367296 23.2 1.609437912 0.120682 0.1618 30 3.401197 24.2 1.643339464 0.125543 0.164911 31 3.433987 25.2 1.676129287 0.130377 0.167908 32 3.465736 26.2 1.707877985 0.135184 0.1708 33 3.496508 27.2 1.738649644 0.139965 0.173594 34 3.526361 28.2 1.768502607 0.14472 0.176295 35 3.555348 29.2 1.797490144 0.149448 0.178909 36 3.583519 30.2 1.825661021 0.15415 0.181442 37 3.610918 31.2 1.853059995 0.158826 0.183898 38 3.637586 32.2 1.879728242 0.163476 0.186281 39 3.663562 33.2 1.905703729 0.1681 0.188596 40 3.688879 34.2 1.931021537 0.172699 0.190845 41 3.713572 35.2 1.955714149 0.177272 0.193033 42 3.73767 36.2 1.979811701 0.181821 0.195163 43 3.7612 37.2 2.003342198 0.186344 0.197237 44 3.78419 38.2 2.026331716 0.190842 0.199259 45 3.806662 39.2 2.048804572 0.195315 0.20123 46 3.828641 40.2 2.070783479 0.199763 0.203153 47 3.850148 41.2 2.092289684 0.204187 0.20503 48 3.871201 42.2 2.113343093 0.208586 0.20686349 3.89182 43.2 2.133962381 0.212961 0.208655 50 3.912023 44.2 2.154165088 0.217312 0.210406(derived from HR for standard Cox proportional hazard model from Table 33)